INFRARED SPECTROSCOPY / Overview 385
See also: Chemiluminescence: Overview; Liquid-Phase.
Ringbom A (1958) Complexation reactions. In: Kolthoff
Fluorescence: Overview. Indicators: Acid Base.
IM and Elving PJ (eds.) Treatise on Analytical Chemistry,
Part I, vol. I, ch. 14. New York: Interscience.
Ringbom A (1963) Complexation in Analytical Chemistry.
New York: Wiley-Interscience.
Further Reading
Schwarzenbach G and Flaschka H (1969) Complexometric
Bishop E (ed.) (1972) Indicators, pp. 209 468, 685 732. Titrations, 2nd edn. London: Methuen.
Oxford: Pergamon. Townshend A, Burns DT, Guilbault GG, et al. (eds.) (1993)
Guilbault GG (1973) Practical Fluorescence Theory, Meth- Dictionary of Analytical Reagents. London: Chapman
ods and Techniques, pp. 397 408, 597 607. New York: and Hall.
Dekker. Welcher FJ (1958) The Analytical Uses of Ethylenedia-
Isacsson UI and Wettermark G (1974) Chemiluminescence minetetraacetic Acid. Princeton, NJ: Van Nostrand.
in analytical chemistry. Analytica Chimica Acta 68: Wilson CL and Wilson DW (1960) (eds.) Comprehensive
339 362. Analytical Chemistry, vol. 1B. Amsterdam: Elsevier.
INDUCTIVELY COUPLED PLASMA
See ATOMIC EMISSION SPECTROMETRY: Inductively Coupled Plasma
INDUCTIVELY COUPLED PLASMA-MASS
SPECTROMETRY
See ATOMIC MASS SPECTROMETRY: Inductively Coupled Plasma
INFRARED SPECTROSCOPY
Contents
Overview
Sample Presentation
Near-Infrared
Photothermal
Industrial Applications
energy of IR photons is thus of the same order of
Overview
magnitude as the energy differences between quan-
tized molecular vibrational states. Transitions be-
P R Griffiths, University of Idaho, Moscow, ID, USA
tween these vibrational modes can be induced by IR
radiation if there is a change in the molecular electric
& 2005, Elsevier Ltd. All Rights Reserved.
dipole moment in the course of the vibrational
This article is a revision of the previous-edition article by
motion. IR spectroscopy is the study of the interac-
B P Clark, pp. 2153 2170, & 1995, Elsevier Ltd.
tion of IR radiation with matter as a function of
photon frequency. This interaction can take the form
of absorption, emission, or reflection. IR spectro-
Introduction
scopy is a fundamental analytical technique for
The infrared (IR) region of the electromagnetic obtaining quantitative and qualitative information
1
spectrum lies between B10 and 12 800 cm . The about a substance in the solid, liquid, or vapor state.
386 INFRARED SPECTROSCOPY / Overview
Table 1 The IR region of the spectrum
1
Region l (cm) n (cm ) n (Hz)
%
4 5
Near-IR 2.5 10  7.8 10 4000 12 800 1.2 1014 3.8 1014
3 4
Mid-IR 5 10  2.5 10 200 4000 6 1012 1.2 1014
3
Far-IR 0.1 5 10 10 200 3 1011 6 1012
It is convenient to divide the IR region into three ratio than dispersive spectroscopy, i.e., measure-
1
parts (Table 1): the far-IR (10 200 cm ); the mid- ments made with prism or grating monochromators.
1
IR (200 4000 cm ); and the near-IR (4000 The following brief description of the principal IR
1
12 800 cm ). (The regions are not exactly defined: techniques  using the near- or mid-IR  illustrates
slightly different boundaries for the IR regions are the range of sample handling possible with IR spec-
found in the literature. In particular, most Fourier troscopy.
transform infrared (FTIR) spectrometers operating in  Diffuse reflection is the term used to describes the
the mid-IR region have a low-wave-number limit of reflection of electromagnetic radiation from a sample
1
400 cm .) Each part of the spectrum plays a dif- after the radiation has undergone multiple scattering
ferent role in analysis according to the different inside a powdered sample or at the surface of a matte
character of the transitions involved in each case. substance. The radiation passes through the  micro-
Mid-IR radiation corresponds to fundamental structural elements of the sample, e.g., the micro-
transitions in which one vibrational mode is excited crystallites of a powder or the surface fibers of a
from its lowest energy state to its first excited state. fabric, and is absorbed in the process before being
For routine analysis a spectrum is normally taken scattered out of the sample to detector. The use of
1
from 400 to 4000 cm . The mid-IR spectrum of a diffuse reflection spectroscopy in the ultraviolet (UV)
substance is effectively a unique fingerprint that can region of the spectrum is a long-established tech-
be used for the purpose of identification by compari- nique. However, until FTIR was established, the
son with a reference spectrum. When no reference weakness of the signal prevented the extension of the
spectrum is available, an IR spectrum can be used to technique into the mid-IR. Diffuse reflection IR
identify the presence of certain structural units that, Fourier transform (DRIFT) spectroscopy has become
irrespective of their molecular environment, give rise a useful technique for obtaining IR spectra from
to characteristic spectral features in a narrow fre- powdered samples (or any matte material) with little
quency range. or no sample preparation.
Near-IR spectroscopy arises from transitions in Photoacoustic IR spectroscopy has similar advan-
which a photon excites a normal mode of vibration tages to DRIFT spectroscopy in its ability to handle
from the ground state to the second or higher excited solids with the minimum of preparation. The prin-
vibrational state (overtones, vide infra) or transitions ciple of this technique is that when a modulated
in which one photon simultaneously excites two or beam of IR radiation is absorbed by a sample, tem-
more vibrational modes (combinations bands, vide perature oscillations set up thermal waves. If the
infra). The use of the near-IR, especially diffuse re- sample is sealed in a cell and surrounded by gas, then
flection spectroscopy, in both quantitative and quali- a microphone can pick up the sound waves in the gas
tative analysis has increased significantly due to and an IR absorption spectrum generated.
better instrumentation and the development of  Specular reflection is the term used to describe
chemometrics to better handle the effect of seriously  mirror-like reflection, from the surface of a sample
overlapping bands. (angle of reflection equals angle of incidence). Spec-
The far-IR region of the spectrum (o200 cm 1) re- ular reflected radiation ostensibly carries no infor-
sults from transitions involving low-frequency torsions mation about the IR absorption of a sample and is a
and internal rotations in liquids and lattice vibrations source of interference in diffuse reflection experi-
in solids and is not commonly used for analysis, al- ments when the sample is not completely matte, i.e.,
though recent developments in instrumentation for has an element of  shininess about it. However,
terahertz spectrometry may change the situation. if the reflected intensity from a sample is due  prin-
The scope and flexibility of IR spectroscopy in the cipally to reflection from the front surface of the
mid-IR region have been greatly increased by the sample, then an absorption index spectrum of the
advent of FTIR spectroscopy. The multiplex and sample can be generated from the reflected intensity
throughput advantages of this technique allow spec- over the whole spectrum using the Kramers Krönig
tra to be run faster and with a greater signal-to-noise transformation. (This complex transformation is an
INFRARED SPECTROSCOPY / Overview 387
integral part of the software packages driving most m1
m2
Massless spring
modern FTIR spectrometers.)
Aqueous solutions have traditionally posed a prob-
lem for IR spectroscopy due to the fact that water is a
r0
strong absorber of IR radiation. This difficulty, for
aqueous solutions and other strongly absorbing liquid
r
and solid samples, can be overcome by using attenu-
ated total reflection spectroscopy. In this technique, Figure 1 Model for a rigid diatomic vibrator.
the phenomenon of total internal reflection is used in
such a way that it is only the evanescent wave as- mass, may be written as
sociated with total internal reflection that enters the
z1 ź z0 þ Dz1 ½1aŠ
1
sample. The evanescent wave penetrates the sample
very short distances only, hence the advantage for
z2 ź z0 þ Dz2 ½1bŠ
2
strongly absorbing species.
FTIR microscopy, in which IR spectra can be ob-
where z0 and z0 are the equilibrium coordinates of
1 2
tained from picogram quantities, is an invaluable
atoms 1 and 2, respectively, and Dz1 and Dz2 are the
nondestructive analytical tool in fields such as foren-
 Cartesian displacements coordinates . An  internal
sic science and pharmaceutical analysis.
coordinate R can be defined as the differences in the
The short data-capture times possible with the
bond length from its equilibrium value
FTIR spectrometer means that time-resolved spec-
R ź Dz2 Dz1 ź r r0 ½2Š
troscopy has become an important means of follow-
ing the course of a chemical reaction in order to
The potential energy of the molecule V(R), which in-
obtain information about kinetics, equilibria, and the
creases as the bond is stretched or compressed, may be
nature of reaction intermediates.
expressed as a power series in the internal coordinate
Matrix isolation IR spectroscopy involves mixing
dVðRÞ
trace amounts of solute into a rare gas matrix at low
VðRÞ ÅºVð0Þ þ R
dR
temperature. The advantage is that the solute mole-
0
! !
cules are isolated so that reactive species can be
1 d2VðRÞ 1 d3VðRÞ
analyzed. Also, the absence of rotational structure
þ R2 þ R3 þ ? ½3Š
2 dR2 6 dR3
and lattice modes increases resolution.
0 0
For chiral molecules a small difference in the
The subscripts (zero) indicate that the derivatives are
magnitude of absorption of left- and right-circularly
to be taken at the equilibrium bound length. Only
polarized IR radiation is observed. This is known as
changes in the potential energy from the equilibrium
vibrational circular dichroism and, since the effect
value are important, so the energy scale may be cho-
can be observed from each normal mode, absolute
sen such that V(0) is zero. Also, the first derivative of
stereochemical information can be obtained from the
the potential energy (dV(R)/dR)0 must be zero by
entire molecule. This is different from the UV analog
definition at the equilibrium position since this cor-
of this effect, where there may often be only one
responds to the energy minimum. This leaves only the
chromophore present.
quadratic and higher terms in the potential function
! !
1 d2VðRÞ 1 d3VðRÞ
VðRÞ Åº R2 þ R3 þ ? ½4Š
The Vibration of Diatomic Molecules
2 dR2 6 dR3
0 0
The Classical Diatomic Rigid Vibrator
The cubic and higher terms are generally small for
An understanding of the nature of vibrational motion
small departures from the equilibrium position. The
is best obtained by first studying a simple system. To
effects of including these contributions in the poten-
introduce some of the basic concepts involved it is
tial energy expression will be considered later. It is,
useful to study the classical diatomic vibrator before
however, a good first approximation to set these
going on to consider the quantum theory.
higher-order terms to zero to give
The simplest model for a diatomic molecule con-
!
sists of two atoms of mass m1 and m2 connected by a 1 d2VðRÞ 1
VðRÞ Åº R2 ź KR2 ½5Š
rigid, massless spring of length r, which has the value
2 dR2 2
0
r0 at equilibrium (Figure 1). If the z-axis is taken to
lie along the internuclear line, then the Cartesian co- A system with a potential function given by eqn [5] is
ordinates of the two atoms, referred to the center of said to be  mechanically harmonic and K is known
388 INFRARED SPECTROSCOPY / Overview
Table 2 The first few Hermite polynomials
as the  force constant for the bond where
!
vHv(x)
d2VðRÞ
K ź ½6Š
dR2 01
0
12x
The harmonic approximation corresponds to a rest- 24x2 2
38x3 12x
oring force F acting on the atoms that is proportional
416x4 48x2 þ 12
to the displacement of the bond length from its equi-
532x5 160x3 þ 120x
librium value (Hooke s law)
F ź KR ½7Š
The vibrational energy levels are given by
With the center of gravity as the coordinate origin, it
1
is straightforward to show that the problem reduces Ev ź v þ hn; v ź 0; 1; 2; y ½13Š
2
to that of a single particle oscillating around the
where
center of mass subject to a harmonic restoring force
whose displacement is equal to the change in the in-
1=2
1 K
ternuclear distance of the molecule, i.e., the internal
n ź
2p m
coordinate R. The mass m of this particle is known as
the reduced mass of the molecule and is given by
Note that this expression is exactly the same as
m1m2
eqn [11], the classical vibration frequency.
m ź ½8Š
m1 þ m2
Vibrational term values G(v) in wave numbers can
Applying Newton s second law gives be defined from eqn [13]:
m d2R
Ev 1 n 1
ź KR ½9Š
GðvÞ Åº ź v þ ź v þ n ½14Š
*
2 dt2
hc 2 c 2
This equation describes the simple harmonic oscilla-
Inspection of eqn [13] shows that the vibrational
tor. The solution is
zero-point energy E0 is given by
R ź R0 si2pnt þ dÞ½10Š
1
E0 ź hn ½15Š
2
which corresponds to simple harmonic motion with
maximum amplitude R0, phase factor d, and fre-
or
quency n given by
1
*
1=2 Gð0Þ Åº n ½16Š
2
1 K
n ź ½11Š
2p m
Figure 2 shows plots of the harmonic diatomic vibra-
It can be shown that the maximum amplitude of tor wave functions.
vibration for each atom is inversely proportional to For harmonic wave function, the properties of the
the atomic mass. Hermite polynomials are such that the selection rules
for vibrational transitions are
The Quantum-Mechanical Harmonic
Dv ź 71 ½17Š
Diatomic Vibrator
The harmonic potential function obtained in eqn [5]
There also exists the gross selection rule that, in
order for electromagnetic radiation to be absorbed,
1
VðRÞ Åº KR2
2
the dipole moment of the molecule must change
during the vibration, which means that a diatomic
can be used in Schrödinger s equation to yield the
molecule must possess a permanent dipole moment
following wave function:
in order to absorb IR radiation. (These selections
ða=pÞ1=4 aR2
rules will be discussed in detail in the section on the
CvðRÞ Åº exp Hvða1=2RÞ ½12Š
2
ð2vv!Þ1=2 intensity of IR transitions.)
Since all energy levels are equally spaced for a
where v is the vibrational quantum number, and
harmonic diatomic vibrator, one line will appear in
the IR absorption spectrum with wave number
ðmKÞ1=2 4p2mn
a ź 2p ź
h h
1=2
1 K
n ź ½18Š
*
where Hvða1=2RÞ are Hermite polynomials (Table 2). 2pc m
INFRARED SPECTROSCOPY / Overview 389
= 3
= 2
re
r
= 1
= 0
r De
re
Figure 3 The Morse potential function.
Figure 2 Harmonic diatomic vibrator wave functions.
Anharmonic Vibrations of a Diatomic Molecule
Table 3 Vibrational frequencies and force constants for select-
ed bonds
The true potential function for a diatomic molecule
1 2 1 a departs from harmonicity, especially for large ampli-
Bond n (cm )K ( 10 Nm )
%
tude vibrations. It is useful to approximate an an-
C H 2960 4.7
harmonic potential using a Morse function (Figure 3)
==C H
3020 5.0
==C H
== 3300 5.9
Eanh ź Def1 exp½ aðr reÞŠg2 ½20Š
(OC) H 2800 4.3
C C 900 2.9
C==C 1650 9.6 where De is the thermodynamic dissociation energy
==
C==C 2050 15
and
C==O 1700 12
==
C==N 2100 17 1=2
m
O H 3600 7.2 a ź o ; o ź 2pn ½21Š
2De
N H 3350 6.2
C F 1100 5.3
C Cl 650 2.2 When this potential energy is used in the Schrödinger
C Br 560 1.9
equation the vibrational levels become
C I 500 1.6
2
a 1 1
1 1 a2_
1 mdyne Šź 100 N m .
GðvÞ Åº v þ ne v þ xene;y; xe ź ½22Š
* *
2 2 2mo
Equation [18] can be used to provide approximate
where ne is the wave number corrected for anhar-
*
values for the force constants of bonds (Table 3).
monicity, and xe is the anharmonicity constant, a
2
Isotopic Substitution
positive number B10 . The negative term in eqn
[22] results in the gap between successive vibrational
If one of the atoms in the diatomic molecule is replaced
energy levels decreasing as v increases. The point at
by another isotopic species, then to a good approxi-
which the levels merge into a continuum corresponds
mation the electronic structure is unchanged. Hence,
to the dissociation of the molecule.
the force constant will be the same for both molecules.
Another effect of anharmonicity is to relax the se-
The change in the vibrational frequency will therefore
lection rules to give
be completely due to the change in the reduced mass. If
m0 and n0 are the reduced mass and wave number of the
*
Dv ź 71; 72; 73; y ½23Š
isotopically substituted molecule, then
Along with the fundamental vibrations,  harmonics
n m0 1=2
*
ź ½19Š
are now observed in the IR spectrum: the second
n0 m
*
harmonic (or first overtone) (v ź 2 v ź 0), third
Similar relationships for frequency shifts induced by harmonic (or second overtone) (v ź 3 v ź 0) and so
isotopic substitution in polyatomic molecules are given on occur with decreasing intensity. A knowledge
later. of the wave numbers of the fundamental and first
Potential energy
Potential energy
390 INFRARED SPECTROSCOPY / Overview
Table 4 Anharmonicity constants for some diatomic molecules enable the use of Newton s equation, the mass-
weighted Cartesian coordinates qi are transformed
1 1
Bond ne (cm )xene (cm )
% %
into a set of new coordinates Qi which results in no
H 12C 2861.6 64.3
cross-terms in the potential function.
H 19F 4138.5 90.07
The Qi are called the  normal coordinates of the
H 16O 3735.2 82.81
system, and this transformation from Cartesian dis-
H 35Cl 2991.0 52.85
placement to normal coordinates is the essence of the
H Br 2649.7 45.21
vibrational problem. Using normal coordinates, the
kinetic and potential energy become
overtone vibrations allows the anharmonicity con-
3N
1X 
stants to be calculated using eqn [22]. Table 4 shows
T ź Q2 ½29Š
i
2
values of anharmonicity constants for a selection of iź1
diatomic molecules.
3N
1X
V ź liQ2 ½30Š
i
2
iź1
Normal Coordinates
Using eqns [29] and [30] in eqn [28] instead of the
Computational and interpretational aspects of vib-
mass-weighted Cartesian coordinates gives
rational spectroscopy are greatly simplified by the
¨
introduction of  normal coordinates . First, mass- Qi þ liQi ź 0; i ź 1; 2; y; 3N ½31Š
weighted Cartesian displacement coordinates for an
These are simply harmonic oscillator equations with
N-atom molecule q1; q2; y; q3N are defined accord-
solutions
ing to
Qi ź Q0 siOlit þ diÞ½32Š
i
q1 ź Om1Dx1; q2 ź Om1Dy1;
q3 ź Om1Dz1; q4 ź Om2Dx2; y ½24Š
with frequencies
1
ni ź Oli ½33Š
In these coordinates the kinetic energy T is given by
2p
3N
1X
Each vibration associated with a normal coordinate
T ź q2 ½25Š

i
2
is known as a  normal vibration . Each atom
iź1
involved in a normal coordinate vibrates in phase
where the dot indicates a time derivate. The potential
with all the other atoms involved in the same vib-
energy is given by
ration. Each atom passes through its equilibrium po-
3N
sition at the same time and reaches each turning
X
1 @V
V ź qiqj
point at the same time.
2 @qi@qj 0
i;jź1
An analysis of the stretching vibrational motion of
3N
X
1
a symmetrical linear triatomic molecule (Figure 4)
ź bijqiqj ½26Š
2
reveals that in terms of mass-weighted Cartesian co-
i;jź1
ordinates q1, q2, and q3 the normal vibrations (ex-
where the bij denote the force constants in the
cluding a zero-frequency translational solution) are
Cartesian displacement coordinate system. (Note
given by
that there are cross-terms in the potential function
1 1
involving two coordinates.)
Qþ ź q1 q2 ½34aŠ
21=2 21=2
The total energy is therefore given by
3N 3N
1X 1X
q q q
M 2m M
E ź q2 þ bijqiqj ½27Š

i
Q ź þ ½34bŠ
2 2
iź1 ijź1
2ðM þ 2mÞ M þ 2m 2ðM þ 2mÞ
1 2 3
If the potential energy term did not include any cross-
with frequencies
terms, then Newton s equation could be applied
1=2
1 K
nþ ź ½35aŠ
d @T @V
2p m
þ ź 0; i ź 1; 2; y; 3N ½28Š
dt @qi @qi

1=2
1 KðM þ 2mÞ
In this case the problem would reduce to the solution
n ź ½35bŠ
2p Mm
of 3N independent equations. Therefore, in order to
INFRARED SPECTROSCOPY / Overview 391
mm Using eqns [22] and [37] the total vibrational and
M
rotational energy is given by
KK
q1 q3 q2
Ev; J 1 1
ź BJð J þ 1Þ þ v þ ne xe v þ ne ½40Š
* *
hc 2 2
Figure 4 The symmetrical linear diatomic molecule.
The selection rules for vibration rotation transitions
The þ and labels have been used to denote that
are the same as for separate transitions:
the two normal vibrations are symmetric and anti-
symmetric stretches, respectively.
Dv ź 71; 72; 73; y
The two bonds in this system are known as  cou-
pled oscillators . This type of system occurs often in
and
the vibrations of polyatomic molecules in which two
oscillators (which can be bonds or groups of bonds)
DJ ź 71 ½41Š
couple to give symmetric and antisymmetric combi-
nations. (In this case, where the system is exactly
Labeling the initial and final levels by double and
symmetrical, the symmetric combination will not be
single primes, respectively, and making the assump-
observed in the IR spectrum because the dipole mo-
tion that the rotational constants for the lower and
ment of the molecule does not change.)
upper vibrational states are the same, then the tran-
sition energies are given by
High-Resolution IR Spectra of
0
nJ ; J00 ź n0 þ Bð J0 J00Þð J0 þ J00 þ 1Þ ½42Š
* *
Linear Molecules
when n0 is the band center (or band origin) given
Each vibrational transition of vapor-phase molecules
by
is accompanied by rotational transitions. In the liq-
uid state the effect of molecular collisions is to
n0 ź neð1 2xeÞ½43aŠ
* *
broaden the rotational lines so that they cannot be
resolved. This is why one observes IR bands as op-
* *
posed to lines in solution spectra. In the vapor phase, n0 ź 2neð1 3xeÞ½43bŠ
however, it is possible to resolve the rotational struc-
ture in a vibrational transition.
n0 ź 3neð1 4xeÞ½43cŠ
* *
The Diatomic Vibrating-Rotator
for the fundamental and first and second overtones,
It is normally a good approximation to express the
respectively. For DJ źþ 1,
total energy due to the motion of the nuclei Enuc as
the sum of the separate energies Evib and Erot (the
00
nJ þ1; J00 ź n0 þ 2Bð J00 þ 1Þ; J00 ź 0; 1; 2; y ½44Š
* *
Born Oppenheimer approximation)
For DJ ź 1,
Enuc ź Evib þ Erot ½36Š
00
nJ 1; J00 ź n0 2Bð J0 þ 1Þ; J0 ź 0; 1; 2; y ½45Š
* *
To a first approximation, wave numbers of the
rotational levels are given by
So it can be seen that the high-resolution spectrum
will consist of two series of lines, one on either side
EJ
ź BJð J þ 1Þ cm 1; J ź 0; 1; 2; y ½37Š
of the band center. The series corresponding to
hc
DJ ź 1 is known as the P branch, and the series
when J is the rotational quantum number and B is corresponding to DJ źþ 1 is known as the R branch.
Note that there is zero intensity at the line center.
the rotational constant given by
Figure 5 shows a schematic diagram of the transi-
h
tions involved and a stick representation of the line
B ź cm 1 ½38Š
8p2Ic
intensities which will be discussed below. Figure 6
shows the high-resolution IR spectrum of the funda-
where I is the moment of inertia of the molecule
mental vibration of carbon monoxide and the same
given by
spectrum of the fundamental vibration of carbon
monoxide and the same spectrum at lower resolution
I ź mr2 ½39Š
e
where only the envelope of the P and R branches can
where re is the equilibrium bond length. be seen.
392 INFRARED SPECTROSCOPY / Overview
5
4
3
Upper
2 vibrational
state
1
J2 = 0
5
4
3
Lower
2
vibrational
state
1
J3 = 0
Frequency
P branch R branch
Position of
Q branch
Figure 5 Schematic diagram of P and R branch transitions for a diatomic molecule, where the line thickness compounds to the
intensity of the transition.
It has been assumed that the rotational constant B are not made, eqn [40] becomes
is the same for the upper and lower vibrational states
Ev; J 1
and that the vibrational terms will be unaffected by
ź BvJð J þ 1Þ þ v þ ne
*
hc 2
the rotational state (interaction between rotation
2
1
and vibration, i.e., a breakdown of the Born
[46]
þ xe v þ ne DvJ2ð J þ 1Þ2 þ ?
2
Oppenheimer approximation). When these assumptions
INFRARED SPECTROSCOPY / Overview 393
R branch P branch
2250 2200 2150 2100 2050
Wave number (cm-1)
Figure 6 The high-resolution infrared spectrum of carbon mon-
oxide showing P and R branches. (Inset on right is the same band
at lower resolution.)
where Bv is the rotational constant associated with
the vibrational level with quantum number v and Dv
is the centrifugal distortion coefficient associated
with the vibrational level v. Centrifugal distortion
Figure 7 Typical PQR branch envelope for the perpendicular
arises due to the fact that a bond will lengthen hence
vibration of a linear polyatomic molecule.
weaken as the molecule rotates. The resultant vibra-
tional rotational energy change is therefore
moment (i.e., are IR allowed) can be classified into
two types: parallel vibrations and perpendicular vib-
0
nv J0; v00J00 ź n0 þ Bv J0ð J0 þ 1Þ Dv J02ð J0 þ 1Þ2
* * 0 0
rations, for which the directions of the changes in the
00 00
Bv J00ð J00 þ 1Þ þDv J002ð J00 þ 1Þ2 ½47Š
molecular dipole moment are parallel and per-
pendicular to the internuclear axis, respectively.
Intensity of Lines in the P and R Branches
The rotational selection rules for parallel vibrations
are the same as for the vibration of a diatomic mole-
The intensity of each rotational line depends on the
cule and one observes P and Q branches as before.
number of molecules occupying the initial rotational
However, for perpendicular vibrations, rotational
state. Using the Boltzmann distribution formula, at
transitions are allowed in which the rotational quan-
thermal equilibrium the ratio of the number of mole-
tum number does not change, i.e.,
cules NJ in rotational state J to the number N0 in the
rotational ground state is given by
DJ ź 0 ½51Š
NJ BhcJð J þ 1Þ
źð2J þ 1Þ exp ½48Š
N0 kT
If rotational constants in both vibrational states are
equal then all Q branch transitions occur at the same
where the (2J þ 1)-fold degeneracy of each rotational
wavelength, the band origin. For a fundamental vib-
state has been taken into account. By differentiating
ration of a perpendicular vibration with unequal ro-
eqn [48] with respect to J and setting the derivative to
tational constants in the two vibrational levels and
zero it can be shown that the maximum population
introducing centrifugal distortion
and hence the maximum intensity line occurs at
0
nv J0; v00J00 ź n0 þðBv Bv ÞJ2 þðBv Bv ÞJ þðDv Dv ÞJ2
* * 0 00 0 00 0 00
1=2
kT 1
0 00
Jmax ź ½49Š þðDv Dv ÞJ; J ź 0; 1; 2; y ½52Š
2Bhc 2
A typical band contour for perpendicular bands is
This corresponds to a maximum intensity at wave
shown in Figure 7.
number nmax given by
*
" 1=2 #
kT 1
* *
nmax ź n072B þ ½50Š
Polyatomic Molecules
2Bhc 2
Internal Coordinates
where the þ and signs refer to the R and P
The set of internal coordinates required to describe
branches, respectively.
the vibrational motion of a general polyatomic mole-
cule consists of the bond stretch r, the bond angle
High-Resolution Vibrational Spectra of
bend f, the out-of-plane (o.o.p.) angle bend g, and
Linear Polyatomic Molecules
the bond torsion t (Figure 8).
The normal vibrations of a linear polyatomic mole- In a polyatomic molecule with 3N 6 vibrational
cule which result in a change in the molecular dipole degrees of freedom (3N 5 for a linear molecule) a set
394 INFRARED SPECTROSCOPY / Overview
"r "
In-phase stretch 2853cm-1 Out-of-phase stretch 2926cm-1
" "
Figure 8 Internal coordinates for polyatomic molecules.
v1
Symmetric stretch
Deformation 1463cm-1 Wag
(3652 cm-1)
v2
Angle bend
(1595 cm-1)
Rock Twist
Figure 10 Characteristic vibrations of the methylene group.
v3
spectra of different molecules within a sufficiently
Antisymmetric stretch
narrow range of frequencies for these bands to be
(3756 cm-1)
used to identify the presence of the structural unit.
These relatively constant bands are known as the
characteristic frequencies or group frequencies of a
molecule. The vibrations associated with methylene
Figure 9 The normal vibrations of H2O.
and methyl groups, which give rise to important
of 3N 6 internal coordinates can be chosen to de- group frequencies, are shown in Figures 10 and 11,
respectively.
scribe the molecular vibrations.
A normal coordinate for a polyatomic molecule
Intensity of IR Transitions
can be expressed as a linear combination of the in-
ternal coordinates. The vibrational behavior of the
Let I0 and I be the incident and transmitted intensi-
atoms can be represented by attaching arrows to
ties, respectively, when infrared radiation passes
show their direction of motion. The lengths of the
through a sample of concentration C and cell length
arrows are in proportion to the maximum ampli-
l. The transmittance T is defined as
tudes of each atom s normal coordinate excursion.
I
The normal vibrations of water (H2O) are shown in
T ź ź 10 aCl ½53Š
I0
Figure 9.
where the quantity a, a function of wavelength, is
Characteristic or group frequencies With a know-
called the absorptivity of the sample. Taking logari-
ledge of atomic masses, the molecular geometry and
thms of the last equation, the absorbance of the
force constants, it is possible to calculate the internal
sample is defined as
coordinate composition of the normal vibrations of
any molecule. Without this mathematical help, it is not I0
A ź log10 ź aCl ½54Š
possible to specify the origin of most of the bands in an
I
IR spectrum which will, in general, contain major
contributions from several internal coordinates. Equation [54] is known as the Beer Lambert law:
However, it is found that certain structural units in the absorbance of a sample is proportional to its
a molecule give rise to bands that appear in the concentration and the cell pathlength. Since A is a
INFRARED SPECTROSCOPY / Overview 395
In-phase stretch 2870 cm-1 Out-of-phase stretch 2960 cm-1
(+ degenerate partner)
Torsion
Out-of-phase deformation 1463 cm-1
(+ degenerate partner)
In-phase deformation 1378 cm-1
Rock (+ degenerate partner)
Figure 11 Characteristic vibrations of the methyl group.
dimensionless quantity, the units of a are the product where aðnÞ has been written to highlight the depend-
*
of the units for reciprocal concentration and recipro- ence of the absorptivity on wave number. Due to the
cal pathlength. For example, with concentration in integration with respect to the wave number the di-
3
mol m and pathlength in meters, the units of a are mensions of the integrated absorptivity are that of
1
m2 mol . When concentration is expressed as mo- the absorptivity divided by the dimension of length.
larity and the pathlength is either in meters of centi- Using eqn [54], the last equation may be expressed as
meters, the absorptivity a is known as the molar
Z
1 I0
absorptivity and is given the symbol e.
A ź log10 dn ½56Š
*
Cl I
The Beer Lambert law is often used in quantitative
band
IR analysis using peakheights in the absorbance
spectrum as values for A, frequently after baseline
While the peakarea is a better measure of band in-
correction. Peakheights, however, are strongly de-
tensity than peakheight in theory, the effect of ab-
pendent on instrumental resolution and an alter-
sorption by neighboring bands leads to baseline
native measure of the absorbance of a band is its
errors that affect the calculation of area more adver-
integrated intensity, which is the intensity integrated
sely than peakheight. It is probably true to say that
over the whole IR band. The integrated intensity may
most contemporary quantitative determinations are
be expressed using the integrated absorptivity A, the
made using peakheight.
absorptivity integrated over the whole IR band, given
The major interaction between the electromagnetic
by
radiation and a molecule is due to the interaction of
the electric field of the former, E, with the dipole
Z
moment of the latter, l. (Magnetic interactions are
A ź aðnÞ dn ½55Š
* *
band much smaller and generally are not important in
396 INFRARED SPECTROSCOPY / Overview
vibrational spectroscopy, although they are respon- and the transition moment becomes
sible for IR circular dichroism.) The interaction gives
hijljf i ź nij egjljeg jnf
rise to a time-dependent perturbation of the quantum
. ź hnijle nf ½61Š
states of a molecule equal to l E. Using the results
of time-dependent perturbation theory (Fermi s
where jnii and jnf are the initial and final vibrational
Golden Rule) it can be shown that the integrated
states, respectively, and le
is the permanent dipole
absorptivity of a transition between initial state hij
moment of the molecule. The permanent dipole mo-
and final state hf j for an isotropic system is given by
ment is now treated as a parametric function of the
NA2p2nif
*
normal coordinates and expressed as a power series
A ź jhijljf ij2 ½57Š
3e0hc lne 10
3N
X6 @l
hnijl0 þ Qp
where hij and hf j are the initial and final states of the
@Qp 0
pź1
transition, l is the molecular dipole moment opera-
3N 3N
X6 X6 @2l
1
tor, NA is Avogadro s number, nif is the wave number
þ QpQr þ ? nf ½62Š
2 @Qp@Qr 0
corresponding to the band center, e0 is the per-
pź1 rź1
mittivity of free space, h is Planck s constant, and c is
the speed of light. The quantity Ignoring the quadratic terms (assuming the system is
Z
electrically harmonic) gives
hijljf i W0lWf dt ½58Š
i
3N
X6 @le
hnijl0 nf þ hnij Qp nf ½63Š
is known as the electric dipole transition moment and e
@Qp 0
pź1
its magnitude determines the intensity of a transition.
The appearance of lne 10 in eqn [57] is to be con-
The total vibrational wave function hnj is given by
sistent with the definition of absorbance using base-
the product of the 3N 6 normal coordinate wave
10 logs. Often the equation will be seen without
functions
divisor, in which case it should be noted that the
absorptivity has been defined using the natural log
hnj ź hv1jhv2jhv3j?hv3N 6j ½64Š
scale.
It should be noted that eqn [57] strictly only ap-
where vp is the vibrational quantum number of the
plies to dilute gases. In condensed phases refractive
pth normal vibration.
index effects become important.
The total vibrational energy is given by
Equation [57] often appears in Gaussian units. The
1 1
Gaussian version is obtained by replacing e0 by 1/4p
Gðv1; v2?v3N 6Þ Åº v1 þ n1 þ v2 þ n2
* *
2 2
to give
1
NA8p3nif þ ? þ v3N 6 þ n3N 6 ½65Š
* *
A ź jhijljf ij2 ½59Š 2
3hc lne 10
Let the vibrational ground state h0j be represented by
The transition electric dipole moment in eqn [57] can
be developed by invoking the Born Oppenheimer
h0j ź h01jh02jh03j?h03N 6j ½66Š
approximation to express the total molecular wave
function as a product of electronic and vibrational
and the state with the pth normal mode in the vp ź 1
parts. (Rotational wave functions do not have to be
state be represented by
included here since eqn [57] refers to an isotropic
1p ź h01jh02jh03j? 1p ?h03N 6j ½67Š
system. That is, the equation is a result of a rota-
tional average which is equivalent to a summation
over all the rotational states involved in the transi- Using the properties of harmonic oscillator wave
tion.) A general molecular state can now be ex- functions (the Hermite polynomials) that
pressed as the product of vibrational and electronic
1=2
h
parts. Assuming that the initial and final electronic
0jQpj1r ź dpr ½68Š
8p2n
states are the ground state jeg ,
where dpr is the Kronecker delta (which is unity if
jii ź jivibi eg ½60aŠ
p ź r and zero otherwise), and
jf i ź jfvibi eg ½60bŠ
0j1p ź 0 ½69Š
INFRARED SPECTROSCOPY / Overview 397
Eqn [63] becomes Because difference bands originate from thermally
populated excited states, they will be more frequently
1=2
h @l observed at lower frequencies and increase in intensity
½70Š
8p2np @Qp 0 as the temperature is raised. (Transitions which occur
from states other than the ground state are known as
hot bands. They are generally weak in mid-IR spectra
Substituting in eqn [57],
at room temperature due to vibrational energy gaps
2
which are relatively large compared to kT).
NA @l
A ź ½71Š
12e0c2 lne 10 @Qp 0
Fermi resonance If an overtone or combination
transition occurs with nearly the same frequency as
a fundamental transition of the same symmetry, then
1=2
@l 12e0c2 lne 10
the anharmonic term in the potential function causes
ź A1=2 ½72Š
@Qp 0 NA
the two vibrations to interact or  mix . This is known
as Fermi resonance. The extent of the mixing increase
Note that the sign of the derivative of the dipole
as the frequency difference decreases. The result is
moment cannot be determined directly by measuring
that the overtone or combination band acquires in-
the integrated absorbance of an IR band.
tensity through having some of the fundamental vib-
Equation [71] shows the origin of the gross selec-
ration mixed into it. Fermi resonance causes the two
tion rule that the dipole moment of a molecule must
bands involved to split apart from the positions they
change in the course of a normal coordinate excur-
would have occupied had no interaction occurred.
sion for the vibration to absorb IR radiation. The
Symmetry of molecular vibrations Every normal
transition moment in eqn [57] is only nonzero for
coordinate of a molecule must transform according
the case where only one vibration is excited and for
to an irreducible representation of the molecular
the situation in which the quantum number of the
vibration involved changes by 71. Hence the selec- point group. If the molecular geometry is known,
then it is a routine matter to use the methods of
tion rule given earlier in eqn [17].
group theory to deduce how many vibrations occur
Anharmonic Effects in Spectra of for each irreducible representation. The procedure is
Polyatomic Molecules
to assign three Cartesian displacement coordinates to
each atom and to use the 3N coordinates as basis
Combination and difference bands Besides over-
functions for a 3N 3N matrix representation of the
tones, anharmonicity also leads to the appearance of
point group. A reducible representation is then ob-
combination bands and difference bands in the IR
tained by taking the trace of these matrices. This
spectrum of a polyatomic molecule. In the harmonic
representation is then reduced to a sum of irreducible
case, only one vibration may be excited at a time (the
representations using
transition dipole moment integral vanishes when the
excited state is given by a product of more than one
1X Ć Ć
Ni ź wiðRÞwredðRÞ½73Š
Hermite polynomial corresponding to different ex-
h
Ć
R
cited vibrations). This restriction is relaxed in the
anharmonic case and one photon can simultaneously where Nl is the number of times that symmetry spe-
Ć
excite two different fundamentals. A weak band ap- cies i occurs, h is the order of the group, wi(R) is the
pears at a frequency approximately equal to the sum character associated with symmetry species i and
Ć Ć
of the fundamentals involved. (Only approximately symmetry operation R, and wred(R) is the character of
because the final state is a new one resulting from the the reducible representation associated with symme-
Ć
anharmonic perturbation to the potential energy try with symmetry operation R.
mixing the two excited state vibrational wave func- One must then remove the irreducible representa-
tions.) tion which result from the three translational and
A difference band is the result of a transition from three rotational degrees of freedom (two for a linear
an excited level of one normal vibration to a higher molecule). These can be identified from the character
energy level of another vibration. The frequency of table of the molecular point group: the translational
the difference band occurs at exactly the difference in degrees of freedom transform as the functions x, y,
the frequencies of direct transitions to the excited and z (denoted Tx, Ty, Tz or x, y, z in the character
states involved from the vibrational ground state. tables); and the rotational degrees of freedom
( Exactly equal in this case because no new vibra- transform as the components of an axial vector
tional state is involved.) (denoted Rx, Ry, and Rz in the character tables). The
398 INFRARED SPECTROSCOPY / Overview
method is illustrated below for a bent triatomic mole- This representation can be reduced using eqn [73]
cule which belongs to the C2v point group. and the information in C2v character table. For ex-
Figure 12A shows the molecule with three Cartesian ample, the number of times NA (the A1 irreducible
1
coordinates associated with each atom. Figure 12B representation) is contained in Gred is given by
shows the effect of C2 rotation on the coordinates. This
1
transformation can be described in matrix form as NA ź ½ð1Þð9Þ þð1Þð 1Þ þð1Þð1Þ þð1Þð3ÞŠ ź 3
1
4
Dx1 Dx3 0 0 0 0 0 0 1 0 0 Dx1
Dy1 Dy3 0 0 0 0 0 0 0 1 0 Dy1
Dz1 Dz3 0 0 0 0 0 0 0 0 1 Dz1
Dx2 Dx2 0 0 0 1 0 0 0 0 0 Dx2
C2 Dy2 ź Dy2 ź 0 0 0 0 1 0 0 0 0 Dy2 ½74Š
Dz2 Dz2 0 0 0 0 0 1 0 0 0 Dz2
Dx3 Dx1 1 0 0 0 0 0 0 0 0 Dx3
Dy3 Dy1 0 1 0 0 0 0 0 0 0 Dy3
Dz3 Dz1 0 0 1 0 0 0 0 0 0 Dz3
From eqn [74] it can be seen that the trace of the C2 The same procedure can be carried out for each
matrix w(C2) is 1. Using the same procedure for the symmetry species of C2v to give
other symmetry operations, the complete reducible
representation formed from the traces of the matrices Gred ź 3A1 þ A2 þ 2B1 þ 3B2 ½75Š
corresponding to all the symmetry operations of the
C2v group is given by From the C2v character table, the three translations
span A1, B1, and B2, and the three rotations span A2,
0
wredðEÞ wredðC2Þ wredðsnÞ wredðsn Þ
B1, and B2. If these are taken away from eqn [75],
Gred 9 1 1 3
this leaves 2A1 and B2. The three normal vibrations
of the bent triatomic molecule, therefore, span these
"z2
irreducible representations. (See Figure 9: the anti-
symmetric stretch is B2.)
This method may be simplified by noting that only
"x2
Cartesian coordinates associated with atoms whose
positions are unchanged by a symmetry operation
"z3
"z1 "y2
may contribute to the trace of a matrix. If a symme-
Ć
try operation R leaves the position of UR atoms un-
Ć
changed, then it can be shown that the character of
"x3
the transformation matrix for not including trans-
"x1
formations and rotations is given by
"y3
"y1
Ć
wðRÞ ÅºðUR 2Þð1 þ 2 cos fÞ½76aŠ
(A)
Ć
"z2
for proper rotations, and
"y2
Ć
wðRÞ ÅºURð 1 þ 2 cos fÞ½76bŠ
Ć
"x2
for improper rotations (reflection, rotation reflection,
"z1
"z3
inversion) where f is the angle through which the
molecule is rotated. (Inversion through the center of
"y1
"y3
symmetry is equivalent to an improper rotation
"x1
"x3 through 1801, a reflection in a plane of symmetry is
an improper rotation through 01, and the identity
(B)
element is a proper rotation through 01.)
For the molecule ClCH3, which belongs to the
Figure 12 (A) Cartesian displacement coordinates of symmetric
triatomic molecules. (B) Cartesian coordinates after C2 rotation.
point group C3v, the number of atoms left unchanged
INFRARED SPECTROSCOPY / Overview 399
by the three symmetry operations I, C3, and sn is very much weaker than chemical bonds, the molec-
ular vibrations are normally very similar to those of
UI ź 5; UC ź 2; Us ź 3 ½77Š
the free molecule. However, the crystal environment
3 n
will generally lower the symmetry of the molecule,
Using eqn [77] in eqn [76] gives:
with the result that the degeneracy of vibrations may
be lifted and vibrations, which were forbidden in the
wðIÞ wðC3Þ wðsnÞ
free molecule can become allowed in the crystal so
Gred 9 0 3
that extra bands can appear. The formal treatment of
the symmetry of vibrations in molecular crystals is
This representation can be reduced using eqn [73] to
obtained by considering the local symmetry in the
give 3A1 þ 3E, Thus, ClCH3 (3N 6 ź 9) has three
crystal unit cell (site group analysis). A more com-
totally symmetric vibrations and three doubly
plete theory, which includes lattice modes, is
degenerate vibrations.
provided by factor group analysis.
Vibrations due to the crystal lattice occur in the
1
Symmetry Selection Rules
far-IR from B50 to 400 cm . It is possible to dis-
tinguish between some molecular and lattice vibra-
Using eqn [61] for a fundamental vibration Qp it can
tions using the fact that molecular vibrations are
be seen that the band intensity is proportional to the
relatively insensitive to the effects of temperature and
following transition dipole moment integral:
pressure while the frequencies of lattice vibrations
0jmaj1p ; a ź x; y; z ½78Š
generally increase with a decrease in temperature and
with an increase in pressure.
Unless the integrand is totally symmetrical the
Infrared Linear Dichroism
integral will be identically zero. Since the vibrational
ground state is always totally symmetric, an IR funda-
For oriented single crystals there will generally be a
mental will only be allowed when one or more com-
difference in the absorption between two linearly
ponents of the dipole moment operator span the
polarized IR beams that are mutually orthogonal and
same irreducible representation as the normal vibra-
orthogonal to the direction of propagations. The
tion. The components of the dipole moment operator
dichroic ratio is defined as
mx, my, and mz span the same irreducible representa-
R
e8ðnÞ dn
* *
tions as the functions x, y, and z, respectively. Hence
band
R
R ź ½79Š
e>ðnÞ dn
* *
a fundamental vibration will only be allowed if it
band
spans the same irreducible representations as x, y, or
z (Tx, Ty, or Tz). where e8 and e> refer to polarization parallel and
It was shown before that a bent triatomic molecule perpendicular to the crystal axis, respectively. If the
undergoes two A1 and one B2 vibrations. Inspection symmetry of the crystal is known, then the dichroic
of the C2v character table revels that z spans A1 and y ratio can give information about the symmetry of the
spans B2. Therefore, all the vibrations are allowed. vibration.
This is not to say that all three bands will appear. The
magnitude of the transitions dipole moments may be
Calculation of Normal Coordinates
so small that a transition may not be observed.
If a molecule has several possible structures which
Given the molecular geometry and a set of force con-
belong to different molecular point groups then the
stants for a polyatomic molecule, it is a routine matter
methods above can be used for structure elucidation,
to calculate the normal coordinates, a procedure
especially in conjunction with the results of the
known as normal coordinate analysis. Suites of com-
analogous analysis for vibrational Raman bands. For
puter programs are readily available that will calculate
example, if a molecule is known to have the molec-
vibrational frequencies and the internal coordinate
ular formula AB4, a group theoretical analysis pre-
composition of each normal vibration. Most of the
dicts that a tetrahedral molecule would have two
early calculation of vibration frequencies were made
active IR fundamentals, whereas a square planar
by Wilson s FG-matrix method, which is briefly sum-
molecule would have three.
marized below. Today, a number of alternative tech-
niques based on semiempirical methods, molecular
mechanics, or density functional theory are also avail-
Infrared Spectroscopy of Crystals
able, in convenient commercial software packages.
In molecular crystals the molecules are held together In the Wilson FG-matrix method, the problem
by van der Waals forces, and since these bonds are is framed in internal coordinates rather than in
400 INFRARED SPECTROSCOPY / Overview
Cartesian displacement coordinates because the force on atomic masses and molecular geometry. It can
constants involved are more meaningful in relation be shown that the vibrational kinetic energy T is
to the chemical structure of the molecule and are given by
more readily transferred between similar molecules.
1
*
 
Also, the theoretical procedure using internal coor- 2T ź RG R ½81Š
dinates are such that the translational and rotational
where a dot denotes the time derivative.
motion of a molecule are automatically taken into
The elements of the G matrix are given in Figure 13.
account.
As an example, the G matrix for a nonlinear tri-
The F Matrix
atomic molecule (Figure 14) is given by
2 3
In the harmonic approximation the potential energy
m3 sin f
m1 þ m3 m3 cos f
6 7
of a molecule can be expressed as r2
6 7
6 7
m3 sin f
7
X
m3 cos f m2 þ m3
G ź6 7
6
Ü
r1
6 7
2V ź FijRiRj ź RFR ½80Š
6 7
4
ij m3 sin f m3 sin f m1 m2 1 1 2 cos f 5
þ þ m3 þ
r2 r1 1 2 1 2 r1r2
r2 r2 r2 r2
where Ri are the internal coordinates, F is the
½82Š
3N 6 3N 6 matrix formed by the force con-
stants, R is a column vector formed by the internal
Ü (The exact from of the elements of G depends on
coordinates and R is its transpose. (In order to have
whether scaled or unscaled coordinates are being
the same dimensions for all coordinates, and there-
used. The above is for unscaled.)
fore all the force constants, the angle bending inter-
nal coordinates are sometimes scaled with a bond
The Secular Equation
length, e.g., in water the angle bend coordinate Da
would become rDa where r is the O H bond length.) The relationship between internal coordinates and
Collectively, the Fij are known as the force field of normal coordinates is defined as
the molecule. These force constants are treated as
R ź LQ ½83Š
empirical parameters whose values are optimized by
obtaining the best fit of calculated to experimental
It can be shown that the matrix vibrational secular
results for vibrational frequencies, Coriolis coupling
equation is given by
constants (which govern a type of coupling between
rotational and vibrational motion), centrifugal dis-
GFL ź LK ½84Š
tortion constants, and mean-square amplitudes of
vibration. The simplest force field neglects all off-
where A is the diagonal eigenvalue matrix and L is
diagonal or interaction force constants. This valence
the matrix of eigenvectors of the matrix product GF.
force field (VFF), in general, gives poor results due to
This last equation is solvable when
the poor number of adjustable parameters (the non-
jGF Elj ź0 ½85Š
zero force constants).
In the generalized valence force field (GVFF) there
where E is the unit matrix and l is a root of
is no neglect of the off-diagonal terms. However, for
the secular polynomial. There will be 3N 6 non-
molecules of any appreciable size the number of force
zero roots, which are equal to the squares of
constants to be determined becomes too large to
vibrational angular frequencies. So the problem is
evaluate them with accuracy. Hence the simplified
essentially the diagonalization of GF, a process which
general valence force field (SGVFF) is frequently used
is easily carried out by computer using numerical
in which all off diagonal force constants are set to
methods.
zero except those involving two internal coordinates
From eqn [83] we have
with common atoms.
The Urey Bradley force field (UBFF) is also com-
Q ź L 1R ½86Š
monly used. This consists of diagonal stretch and bend
force constants together with repulsive force constants 1
so that the elements of L can be used to obtain a
representing nonbonded atom atom interaction.
picture of the normal vibration in terms of the in-
ternal coordinates. However, the reported results of a
The G Matrix
normal coordinate analysis often include the poten-
The kinetic energy part of the vibrational problem tial energy distribution (PED) for this information.
is expressed in the G matrix whose elements depend The PED is the fraction of the potential energy
INFRARED SPECTROSCOPY / Overview 401
rr
G2 2 1 G3 3 2 1 G2 µ1 + µ2
rr
G1 µ1c
rr
G2 - p23µ2s
r
2 3
G1 (1) p13µ1s c
r 2
G1 1
G2 1 2 1 1 s 213cÈ234 + p14s 214cÈ243)µ1
rr (1)
(1) - (p13
µ1+ µ3 + ( +
p2 p2 p2 p2
G3 12 23 12 23- 2p12p23c )µ2

3 4
(p12cÈ314)µ1
G2 1 2 + [(p12 - p23cĆ123 - p24cĆ124)p12cÈ314 +
(1)

3
(s 123s 124s2È314 + c 324cÈ314)p23p24]µ2
G2 2 1
r
G2 0 2 1
(1)
(1)- p12s [(p12 - p12c 1)µ1 + (p12 - p23c 2)µ2]
0
3
G2 2 - (s 25s 34 + c 25c 34cĆ1)p12p14µ1
(2)

2
4
p12µ1
(2) [(s 214c 415c 34 - s 1c 35)p14 + (s 215c 415c 35 - s 214c 34)p15] ×
1
3
s 415
1
G1 (1)
r 2
(1)[c 415 - c 314c 315 - c 214c 215 + c 213c 214c 315)p12p13
1
2
3
+ (c 413 - c 514c 513 - c 214c 213 + c 215c 214c 513)p12p15
1
G1 2
(2)
+ (c 215 - c 312c 315 - c 412c 415 + c 413c 412c 315)p14p13

4
µ1
+ (c 213 - c 512c 513 - c 412c 413 + c 415c 412c 513)p14p15] ×
4
s 214s 315
2
5
G1 1 1
(1)
r
c - c c
2 4
cÈ =

3 4
s s
Å‚
G1 1 1
(1)
3


(e × e ).(e × e )
3 5
2
c =
s s e e
,
e

G1 1 1
(2)



4 5
Figure 13 Representation of common elements of the G matrix. Key: atoms common to both coordinates are double circles in
horizontal line; number of common atoms as superscript; atoms above horizontal belong to first coordinate; those below belong
n
second; n and m are numbers of (noncommon) atoms above horizontal on left and below horizontal on left, respectively.
m
c, cos; s, sin; rab ź 1/rab; ma ź 1/mass a.
The sum of the diagonal elements in the PED can
3
exceed 100% due to the neglect of the off-diagonal
contributions.
r1 r2

Use of Symmetry
When a molecule possesses symmetry the vibrational
problem may be simplified by transforming the in-
1 2 ternal coordinates to symmetry coordinates. A vib-
ration of a certain symmetry will be composed solely
of symmetry coordinates belonging to the same sym-
Figure 14 Nonlinear triatomic molecule.
metry species.
The transformation to symmetry coordinates RS is
of a normal mode contributed by each force constant
given by
Fij. The diagonal elements of this distribution for
the major components of a normal vibration are
RS ź UR ½88Š
quoted as a percentage. For vibration Qp and
internal coordinate Ri
The symmetry coordinates and hence the coeffici-
!
ents of the (unitary) symmetrization matrix
100FiiL2
ip
are obtained by applying symmetry projection
P
PED ðRiÞ Åº % ½87Š
FiiLip
i operators to the internal coordinates. For irreducible
402 INFRARED SPECTROSCOPY / Sample Presentation
representation i translations belonging to the symmetry species in
X
Ć Ć question (which can be deduced using the methods
RS ź N wiðRÞR R ½89Š
i
Ć described in the section on symmetry); Ix, Iy, and
R
Iz are moments of inertia about the three Cartesian
where N is a normalization factor.
axes; rx, ry, and rz are 1 if the respective rotation
A similarity transformation using U is carried out
belongs to the symmetry species concerned and 0
on the F and G matrices
otherwise; m is the mass of an atom which is a
member of a set of symmetrically equivalent atoms
FS ź UFh
½90aŠ
and a is the number of external (rotational and
translational) symmetry coordinates which these at-
GS ź UGh
½90bŠ
oms give rise to. The product on the right-hand side
The symmetrization process produces the block-
involves all sets of symmetrically equivalent atoms.
factored matrices FS GS. Hence the product GSFS
will be block-factored and each block may be
See also: Chemometrics and Statistics: Optimization
diagonalized separately.
Strategies. Chiroptical Analysis. Fourier Transform
Techniques. Infrared Spectroscopy: Sample Presenta-
Isotopic Substitution in Polyatomic Molecules: The
tion; Near-Infrared. Photoacoustic Spectroscopy.
Teller Redlich Product Rule
Observation of the changes in frequency that occur
Further Reading
in a vibrational spectrum as a result of isotopic sub-
Bellamy LJ (1970) The Infrared Spectra of Complex
stitution of one or more atoms is an important meth-
Molecules, 3rd edn., vol. 1. London: Chapman and
od for assessing the accuracy of molecular force
Hall.
fields. Isotopic substitution is also important for
Bellamy LJ (1980) The Infrared Spectra of Complex Mole-
making band assignments in large molecules: the on-
cules, 2nd edn., vol. 2. London: Chapman and Hall.
ly vibrations to be shifted will be those involving the
Bright Wilson E Jr, Decins JC, and Cross PC (1955)
isotopically substituted atoms.
Molecular Vibrations. New York: McGraw-Hill.
For isotopic substitution in which the molecular
Chalmers JM and Griffiths PR (eds.) (2002) Handbook of
point group is unchanged, the Teller Redlich product
Vibrational Spectroscopy. Chichester: Wiley.
rule links the two sets of vibrational frequencies.
Colthup NB, Daley LH, and Wilberley SF (1990) Intro-
There is one product rule for each symmetry species duction to Infrared and Raman Spectroscopy, 3rd edn.
of the molecule as follows London: Academic Press.
Herzberg G (1945) Infra-red and Raman Spectra of Poly-
( t !r )1=2
y
atomic Molecules. New York: Van Nostrand.
Pn M Ix rx Iy Iz rz m0 a
ź P ½91Š
Nakamoto N (1978) Infrared and Raman Spectra of
0 0 0
Pn0 M0 Ix Iy Iz m
Inorganic and Coordination Compounds, 3rd edn.
New York: Wiley.
where a prime is used to distinguish properties of the
Schrader B (ed.) (1995) Infrared and Raman Spectroscopy.
two molecules and P denotes a product. The prod-
New York: VCH.
ucts on the left-hand side include all vibrations in the
Woodward LA (1972) Introduction to the Theory of
particular symmetry species to which the equation
Molecular Vibrations and Vibrational Spectroscopy.
applies. M is the molecular mass; t is the number of Oxford: Oxford University Press.
Sample Presentation
J Chalmers, VSConsulting, Stokesley, UK
extremely important analytical technique. Mid-
infrared spectroscopy is used extensively in applica-
& 2005, Elsevier Ltd. All Rights Reserved.
tions involving qualitative analysis, providing either
functional group or structural information about a
sample or fingerprinting (identifying) a material.
There is also widespread use of the technique for
Introduction
quantitative purposes, since the absorbance of a band
is proportional to the concentration of the species
Since an infrared spectrum can be recorded from
that gives rise to the absorption band.
almost any material, infrared spectroscopy is an